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Sufficient Set of Integrability Conditions of an Orthonomic System

  • Michal Marvan
Article

Abstract

Every orthonomic system of partial differential equations is known to possess a finite number of integrability conditions sufficient to ensure the validity of them all. Here we show that a redundancy-free sufficient set of integrability conditions can be constructed in a time proportional to the number of equations cubed.

Keywords

Orthonomic system Integrability conditions Monomial ideal 

Mathematics Subject Classification (2000)

35N10 12H20 52B20 

References

  1. 1.
    J. Apel, R. Hemmecke, Detecting unnecessary reductions in an involutive basis computation, J. Symb. Comput. 40, 1131–1149 (2005). zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    G. Birkhoff, Lattice Theory, 3rd edn. (Am. Math. Soc., Providence, 1967). zbMATHGoogle Scholar
  3. 3.
    A.V. Bocharov, V.N. Chetverikov, S.V. Duzhin, N.G. Khor’kova, I.S. Krasil’shchik, A.V. Samokhin, Yu.N. Torkhov, A.M. Verbovetsky, A.M. Vinogradov, Symmetries and Conservation Laws for Differential Equations of Mathematical Physics. Translations of Mathematical Monographs, vol. 182 (Am. Math. Soc., Providence, 1999). Google Scholar
  4. 4.
    F. Boulier, An optimization of Seidenberg’s elimination algorithm in differential algebra, Math. Comput. Simul. 42, 439–448 (1996). zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    B. Buchberger, Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal, Ph.D. Thesis, Univ-Innsbruck, 1965. Google Scholar
  6. 6.
    B. Buchberger, A criterion for detecting unnecessary reductions in the construction of Gröbner bases, in Proc. EUROSAM ’79, ed. by E.W. Ng. Lecture Notes in Computer Science, vol. 72 (Springer, Berlin, 1979), pp. 3–21. Google Scholar
  7. 7.
    M. Caboara, M. Kreuzer, L. Robbiano, Efficiently computing minimal sets of critical pairs, J. Symb. Comput. 38, 1169–1190 (2004). CrossRefMathSciNetGoogle Scholar
  8. 8.
    J.-C. Faugère, A new efficient algorithm for computing Gröbner bases (F4), J. Pure Appl. Algebra 139, 61–88 (1999). zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    R. Gebauer, M. Möller, On an instalation of Buchberger’s algorithm, J. Symb. Comput. 6, 275–286 (1988). zbMATHCrossRefGoogle Scholar
  10. 10.
    V.P. Gerdt, Gröbner bases and involutive methods for algebraic and differential equations, Math. Comput. Model. 25, 75–90 (1997). zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    V.P. Gerdt, Yu.A. Blinkov, Involutive bases of polynomial ideals, Math. Comput. Simul. 45, 519–541 (1998). zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    G. Grätzer, General Lattice Theory (Akademie, Berlin, 1978). Google Scholar
  13. 13.
    W. Hereman, Review of symbolic software for Lie symmetry analysis, Math. Comput. Model. 25, 115–132 (1997). zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    E. Hubert, Notes on triangular sets and triangulation-decomposition algorithms II: Differential systems, in Symbolic and Numerical Scientific Computation, Proc. Conf. Hagenberg, ed. by F. Winkler, U. Langer, Austria, 2001. Lecture Notes in Computer Science, vol. 2630 (Springer, Berlin, 2003), pp. 40–87. CrossRefGoogle Scholar
  15. 15.
    M. Janet, Leçons sur les Systèmes d’Èquations aux Derivées Partielles (Gauthier-Villars, Paris, 1929). zbMATHGoogle Scholar
  16. 16.
    R.M. Karp, R.E. Tarjan, Linear expected-time algorithms for connectivity problems, J. Algorithms 1, 374–393 (1980). zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    B. Kruglikov, V. Lychagin, Mayer brackets and solvability of PDEs. I, Differ. Geom. Appl. 17, 251–272 (2002). zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    B. Kruglikov, V. Lychagin, Mayer brackets and solvability of PDEs. II, Trans. Am. Math. Soc. 358, 1077–1103 (2006). zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    W. Küchlin, A confluence criterion based on the generalised Newman lemma, in EUROCAL ’85, Proc. Conf. Linz, Vol. 2, ed. by B.F. Caviness, April 1–3, 1985. Lecture Notes in Comput. Sci., vol. 204 (Springer, Berlin, 1985), pp. 390–399. Google Scholar
  20. 20.
    M. Marvan, Sufficient set of integrability conditions of an orthonomic system: Extended abstract, in Global Integrability of Field Theories, Proc. GIFT 2006, Cockcroft Inst., Daresbury, ed. by J. Calmet, W.M. Seiler, R.W. Tucker, November 1–3, 2006 (Universitätsverlag, Karlsruhe, 2006), pp. 243–247. http://www.uvka.de/univerlag/volltexte/2006/164/pdf/Tagungsband_GIFT.pdf. Google Scholar
  21. 21.
    E. Miller, B. Sturmfels, Combinatorial Commutative Algebra, Springer Graduate Texts in Math., vol. 227 (Springer, New York, 2004). Google Scholar
  22. 22.
    P.J. Olver, Equivalence, Invariants, and Symmetry (Cambridge University Press, Cambridge, 1999). Google Scholar
  23. 23.
    J.F. Pommaret, Systems of Partial Differential Equations and Lie Pseudogroups (Gordon and Breach, New York, 1978). zbMATHGoogle Scholar
  24. 24.
    G.J. Reid, Algorithms for reducing a system of PDEs to standard form, determining the dimension of its solution space and calculating its Taylor series solution, Eur. J. Appl. Math. 2, 293–318 (1991). zbMATHCrossRefGoogle Scholar
  25. 25.
    G.J. Reid, P. Lin, A.D. Wittkopf, Differential elimination-completion algorithms for DAE and PDAE, Stud. Appl. Math. 106, 1–45 (2001). zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    G.J. Reid, A.D. Wittkopf, A. Boulton, Reduction of systems of nonlinear partial differential equations to simplified involutive forms, Eur. J. Appl. Math. 7, 604–635 (1996). CrossRefMathSciNetGoogle Scholar
  27. 27.
    C. Riquier, Les Systèmes d’Èquations aux Derivées Partielles (Gauthier-Villars, Paris, 1910). Google Scholar
  28. 28.
    C.J. Rust, Rankings of derivatives for elimination algorithms, and formal solvability of analytic PDE, Ph.D. Thesis, University of Chicago, 1998. Google Scholar
  29. 29.
    C.J. Rust, G.J. Reid, Rankings of partial derivatives, in Proc. ISSAC 1997 (ACM, New York, 1997), pp. 9–16. CrossRefGoogle Scholar
  30. 30.
    C.J. Rust, G.J. Reid, A.D. Wittkopf, Existence and uniqueness theorems for formal power series solutions of analytic differential systems, in Proc. ISSAC 1999 (ACM, New York, 1999), pp. 105–112. CrossRefGoogle Scholar
  31. 31.
    D.J. Saunders, The Geometry of Jet Bundles, London Math. Soc. Lect. Notes Series, vol. 142, (Cambridge Univ. Press, Cambridge, 1989–1929). zbMATHGoogle Scholar
  32. 32.
    J.M. Thomas, Riquier’s existence theorems, Ann. Math. 30, 285–310 (1928). CrossRefGoogle Scholar
  33. 33.
    A. Tresse, Sur les invariants différentiels des groupes de transformations, Acta Math. 18, 1–88 (1894). CrossRefMathSciNetGoogle Scholar
  34. 34.
    A.M. Verbovetsky, I.S. Krasil’shchik, P. Kersten, M. Marvan, Homological Methods in Geometry of Partial Differential Equations (MCCME, Moscow, under preparation) (in Russian). Google Scholar
  35. 35.
    F. Winkler, Reducing the complexity of the Knuth–Bendix completion algorithm: a “unification” of different approaches, in EUROCAL ’85, Proc. Conf. Linz, Vol. 2, ed. by B.F. Caviness, April 1–3, 1985. Lecture Notes in Comput. Sci., vol. 204 (Springer, Berlin, 1985), pp. 378–389. Google Scholar
  36. 36.
    F. Winkler, B. Buchberger, A criterion for eliminating unnecessary reductions in the Knuth–Bendix algorithm, in Algebra, Combinatorics and Logic in Computer Science, Vol. II, ed. by J. Demetrovics et al. Coll. Math. Soc. J. Bolyai, vol. 42 (J. Bolyai Math. Soc., North-Holland, 1986), pp. 849–869. Google Scholar
  37. 37.
    F. Winkler, B. Buchberger, F. Lichtenberger, H. Rolletschek, Algorithm 628: an algorithm for constructing canonical bases of polynomial ideals, ACM Trans. Math. Softw. 11(1), 66–78 (1985). zbMATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    A.D. Wittkopf, Algorithms and implementations for differential elimination, Ph.D. Thesis, Simon Fraser Univ., 2004. Google Scholar
  39. 39.
    T. Wolf, The integration of systems of linear PDEs using conservation laws of syzygies, J. Symb. Comput. 35, 499–526 (2003). zbMATHCrossRefGoogle Scholar

Copyright information

© SFoCM 2008

Authors and Affiliations

  1. 1.Mathematical Institute in OpavaSilesian University in OpavaOpavaCzech Republic

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